CHAP, ill] PARTITIONS. 115 



Also, 



(- l)'C(n - y) = 1 or (y = 3z 2 =F 2), 



j,=0 



according as n is or is not of the form z(z + l)/2. 



A. De Morgan 28 considered the number u x , v of ways x can be formed 

 additively from y and numbers ^ #. Adding y to each such composition 

 of x y, we see that 



U x , y = U x -y, 1 + Ux-y. 2 + ' ' ' + U x -y, y. 



Subtracting from this the equation obtained by decreasing x and y by unity, 

 we get 



(b) U x , y U x 1, yl = U x y, y 



Regard y as fixed and the second u as a given function, we have a difference 

 equation of order y whose general integral is of the form 



U X , y = Ay-l + A a2 P 2 + ' ' + A a Py, 



where A an is a rational integral function whose degree a n is the greatest 

 integer in (n y)fy, while P n is a circulating function with a cycle of n 

 values. In particular, 



Wx,2=2~4~t'4(~ l) x > 



U., 3 = 4* {6z 2 - 7 - 9(- 1) + 8(0* + 7") }, 



U., 4 = sir {6z 3 + 18z 2 - 27x - 39 + 27(z + 1)(- 1)* 



+ 32(/3 z - 1 + 7*- 1 - P x - T Z ) + 54i* + 54(- i) x }, 



where /3, 7 are the imaginary cube roots of unity and i = V 1. Thus 

 , 3 = x\ x* - 1, x 2 - 4, z 2 + 3, x 2 - 4, x 2 - 1, 



according as x = 0, 1, 2, 3, 4, 5 (mod 6). Similarly, u x , 4 has 12 forms 

 depending on the residues of x modulo 12. Again, u x , s is the integer 

 nearest z 2 /12, and u x , 4 that nearest to (x 3 + 3z 2 )/144 or (x 3 + 3z 2 - 9x)/144, 

 according as x is even or odd. 



A. Cauchy 29 proved (3) and the other formulas of Euler 3 and the related 

 ones involving a finite number of factors : 





= 1 I 1 '^x I d -*")(*- *") 



j=o I t (1 - 



. i 4. 



~ - 1 ~r 



I ^ y \" /\" / 3 | 



~T /-j /W1 -ION /- J5\ "^ 



(1 - *")(! ~ 



P(-x) l-t 



(i - 0(1 - 2 )(i - i 3 ) 



28 Cambridge Math. Jour., 4, 1843, 87-90. 



"Comptea Rendus Paris, 17, 1843, 523; Oeuvres, (1), VIII, 42-50. 



